Gearing



j I i Q INVENTOR ,1933. E. WILDHABER ,3

GEARING Filed FeB. 17, 1950 3 Sheets-Sheet 1 Jan. 24, 1933. v w 1,895,360

GEARING Filed Feb. 1?, 1950 s Sheets-Sheet 2 FIG-9 l NVENTOR 1933- E. WILDHABER GEARING Filed Feb. 17. 1930 3 Sheets-Sheet 3 INVENTOR Patented Jan. 24, 1933 UNITED: STATE nnnnsr wILnHAnEn, or BROOKLYN, New YORK f i i ennui;

Applicationfiled Iebruary 17, 193d Serial No; 429,092

Thev present invention relates to gearing,

namely to the tooth shape .of gears mounted numbers of teeth.

A further aim of the "present inventionisv to providegear pairs of any'ratio havingteeth of great strength and operating with in creased quietness. A further objectof the present invention is to provide a tooth'form which. tends to wea r more evenly than the conventional forms of tooth, and whichis therefore more apt toconserve the correct tooth action and quiet operation.

Another objectv of the present invention is to devise an improved form oftooth which can be produced withhigh accuracy as well as expediency, and which maybe finished after hardening aswell as the conventional involute teeth. i

. A still further aim' is to provide gear pairs with improved tooth shape of such character that each member of the pair is conjugate to the same third member or gear, and to provide a system of interchangeable,gears, of improved tooth form. i

More particularly an improved tooth form for spur and helical gears shall be provided,

which may be produced with high accuracy and expediency, and in which both members contain curved tooth profiles] J L The tooth shape of gears] of the last said character hitherto has been based on the tooth form of a rack. The tooth formof the rack was ordinarily given the simplest profile of the gear system. Such rack has been gener ally referred to as a basic rack.

In accordance with the present invention I depart from this practice and embody the simplest tooth form of the gear system on a pinion, that is to say on an external gear with a comparatively small numberof teeth. How this may be done, and what relations should be observed that this may be accomplished will be fully described hereafter.

A fu'rtherobject of the present invention is to provide a system of spur gears or helical Another aim in view is 'to devise a {Q gears of which eachisconjugate-to' the same 69 gears derived from a basic pinion ofsimplest j tooth form and of'suclr character that'each i V I gear of a gear pair is conjugate to thefsame' pinion. Moreover gear pairs having a 'ratio between one to one and twoto one shall be provided, in which both gears of fthe'ipair containwholly convex toothprofiles'conju i H gate to the same basic pinion. p a

basic pinion, which contains tooth profiles" of symmetrical form with respect to a profile normal erected at amean profilepoint, and

further to devise a pinion of'lessthan 25 w teeth having a convex tooth profile whose curvature radii area minimumat a point intermediate the profile ends andwhose curvature radiiincrease at' points "disposed nearer to the tip end as well as at points disposed? nearer to the root end of the teeth. I Further an'improved system of spur and I helical gearing shall be devised, which is specifically adapted to'produ ction with a gear s aper cutter. Another object is to estab lish an improved system of spur and helical r v gearing based on a pinion of the size of a gear shaper cutter, which has substantiallyf sym metrical tooth profiles witha minimum curvature radius at a of a tooth profile. C

Various further objects will appear in the course of the specification-and from recital of the appended claims. f

My invention is exemplified in the accomjg p-anying drawings, inwhich j 'j I Fig. 1 is a diagram explanatory of 5 one of the principles underlying the present in.- vention, and a partial view'of apair of gears containing tooth profilesxof a character re- 5'95 :ferredto in the specification. Fig. Q'is'a diagrammatic view of th profile I of a pinion formed in accordancewith the H present invention and suited to constitute the basic'tooth-form of an improved pairgof 9 5 gears l l a F 1g; 3 is a diagrammaticview'similar to Fig. 2 further explanatory of a preferred form of profile of a pinion .or a basic pinion 5 and affordinga comparison between" said 100' TENT point intermediate the ends i v of a device for finishing gears constructed profile and the involute profile of a conventional pinion.

Fig. 4 is a partial view of a pair of spur gears constructed in accordance with the present invention.

Fig. 5 is a partial view, partly in section,

in accordance with the present invention'by I abrasion, The novel feature resides in the rotary motion of the blank and in the 'speinterrelationbetween the translatory and the cific means for securing such novel interrela tion, as will be fully explained hereafter."

. I 'Fig. 6 and Fig. 7 are diagrammatic views of two different embodiments for securing the proper interrelation between the transla- 7 toryrnotion of thefblanlcrelativelytoan abrasive wheel and the angular .motion I about the axis 015 the blank.

. ig; 8 is afdiagram usefulfurther ex plaining the relation of translatoryv and rotary motion between a gear blank and a tool.

Fig. 9 is a diagramexplanatoryof a simple way of obtaining a suitable interrelation of translatory-and rotary motion between a blank and a tool, for accurately forming pinare diagrams similar to Fig. .9 and explanatory of a preferred p-ro- 5mg. 10 and Fig.

cedure n whichfboth a translatory motion and a rotary motion are performed by the blank. 5

Fig. 12 isa view of a circular cam cooperating with a convex stationary element, illus trative of'one embodiment for eifecting the interrelation between rotary motion and translatory motion ofthe blank, asindicated Y in diagrams Fig. 10 and Fig. 11.

Fig; 13 and Fig. 14; are views of different positions of 'a circular cam cooperating with a concave stationary element, illustrative of another and often preferred embodiment for 'efiecting an interrelation between rotary motion and translatory motion of the blank, as indicated in diagrams Fig. 1-0 and Fig. 11.

Fig. 15'is partly a view of an adjustable camand a stationary element cooperating therewith, and partly a section along lines 15--15 of Fig.'16. v

. Fig. 16 is a front elevational view, partly I insection, of an adjustable cam of symmetrical form cooperating with an adjustable stationary element of. concave profile; The cam andstationary element shown in Fig. 15 and Fig, 16 are adapted to effect the desired motions for forming contours shaped in acvention. V

cordan'ce with the present invention.

. Fig. 17 is a view of a hollow;camand a stationary element of convex involute profile, illustrative of a modified way of obtaining contours in accordance with the presentin- ,Fig. 18 and Fig. 19. are viewsof adjustable c rcu ar cams cooperating with stationary onsconstructed in accordance with the present invention;

elements, showing said cams and stationary elements in adjusted positions corresponding to the production ofgears. of difi'erent diameter and constructed in accordance with the present invention.

' In Fig. 1 the numerals l1 and 12' denote two equal gears meshing with each other and rotatable on parallel axes l3 and ltrespectively. The gears contain identical teeth 15, 16 having tooth profiles "17, 18. The pitch circles 19, 20 are shown to contact with each other in a point 21 hereafter referredto as the pitchpoint. A pair of tooth profiles is shown enlarged in a position 17" and 18? respectively, in which contact is made at said pitch'p'oint 21. I

. I hall now introduce the pr sent inven; tion gradually and first assume that the profiles 17,"18f are singlecircular arcs,'centered atfpoints' 2 2, 23.. Points 22,23 are alsol the projections of the gear. centers-18, 14. to the joint profile normal 245at pitch point 2L 1 Tooth profiles l'i, l8;o f exactly circular I V I 7 form will now be demonstrated to fall very little short of transmitting trueuniif-ornrmotion, and to be acceptable in many cases, espe cially when the number of teeth of the gears 11 and 12 is comparatively large; It will then further be shown that a slightmodification of the circular profilesl is. suificient to provide acorrection which makes thehg ear.

profiles accurate, so that true uniform motion may be transmitted by the gears. ,This correction, it will be. seen, will keep the toothprofiles in simple form, which can be easily reproduced, and which byreaso'nlof its siniplicity is suited to constitute a-basic form-of tooth, as will be further described.

The circular tooth profiles are, also shown in dotted lines 17 18 in a position, which corresponds to a different angular position of gears 11, 12. In this latter position of the gears, the profilecenters occupy positions 22 and 23respectively.' r a I i The angles 221322 and 23.14..23

are seen to represent the turning anglesof the. gears, as compared .with the position where contact is made at pitch point 21, andare assumed to be exactly alike. It is understood thatcorresponding turning angles of the gears 11' and-12 must be exactly alike,;if

the tooth profiles are mathematically accur rate. On the other handifthe tooth profiles fall short of being mathematically accurate,

and the gears are turned byexactly equal trated, namely by a-distance equal to the in I crease in length of distance 2223f .as com;- 1

the turning angle of thegears.-;Angleu is .seen to be the same as angle;22-1-3 22/ and an le 2314 23 above referred to. I adius 1322,.and also,radius.1 l-.-23rare equal to R'cos a, and distance-2223 equals 2R-sin a It can-be readily demonstrated that the projection of distance 22" 23" to normal 24 is equal to :distance 22-23 and therefore equal to 2R-sin a."

a Further it is seen that the normalidist ance of center23 and of c'enter 22ft from normal 24. or its elongation equals each Rcos .a;( 1-

cos u), and. that distance 22",-,-28.-can'be resolved 3 into a; distance 2R-sin' a and V a distance 2R-cos' a-(l-cos a) which are perpendicular to each other. Hence it is obvious that distance 22"2 3" may beexpressed as thesquare root of the squares of said two distances which are perpendicular to each other. In mathematical terms Distance 22-23' /(2 R-sin a a-[213. g (1 5 =2R-sin a g/lwtT -Ti m y J1 j TotalerrorE. J A =distance 22" 23 distance'22'23 =2R-sin a[ /1 +[cotana- (1 cos 11.)]. ll

For turning angles uvwhich are comparatively small, and notmuch exceeding sayten or fifteen degrees or the expression within parentheses may betransformed to the closely approximate and simple'r expression j 2tan a. 7 so thatthe total error amountsto approximately: 1 1

cos a Ei tan a 1-. cos

Herein the factor v cos a is used in place of the equal factor, h

' sine A i an? a." a

which is directly obtained above. A 7 Error E is understoodto be the tot-aler-ror composed of the added errors of both profiles. 'i

In an. example R=1.500';ia=20 degrees;

and turning angle u=10 degrees, the total error E may be computed as E' ==,.000,897 or an error, per gear of .00045., A maximum is acceptable in many instances. The. error is zero at zero turning angle and increases to the above maximumerror at a turningangle of 10 degrees in either-direction tromazero turnmg angle; Truly circular profiles of the character and dimensions referred to, are

member isconjugate to a pinion havingcin cular tooth; profiles as described will con tam errors otthe above sald character, al-

though the tooth profiles of said are dlfierent from circular arcs. I

';.';It is thereforein manyinstances-possible gears usually therefore suited-to turnthe gear-s11 and 12 I 7 through anarc of degrees with a max1 mnm error as given above.

-s'o Furthermore a pair of gears, of which each to base a system of gearing on a pinion havling circular-tooth profi-les,:and' to make each gear ofia 7 pair of gears conjugate to-said pin- 1on.

; f f Moreover, inaccordance with the preferred embodiments of, thepresent invention a correction is. provided, which reduces or elim- :inates the small error existingon gears based on-apinion having circular tooth profiles of the character described, and which makes'it be further explained hereaften,

Reference is made to F ig. 2, Where a circular.

possible toobtain alarger arc of action with V l less error or with no'jerror. AdVantages alreadyexisting are thereby amplified,as will tooth profile 18 is-indicated in dotted lines,

and in a largerscale than was used in Fig. 1. The center23 of said circular profile is again situatedonthe projection of thegear center to profile normal 2 1; at 'pitchpoint 21. Circular tooth profiles (l8).-pro,v-ided on a-Jpair of equal gears ceaseto contact with each other, When said gears are turned through equal angles, as has beenexplained With reference to Fig. 1. To; maintain-tooth contact in various angular positions of such gears, tooth profiles may be provided, which extend outside of said circular arc, on both sides of pitch point 21. A toothprofile of this character is denoted with numeral'27in Fig. 2.

"This profile has'ai minimum curvature radius at'pitch POl IIt 21 and gradually lncrea'slng curvature radii at profilepoints disposed on each side of pitch point 21. A curvature radius is understood to be the radius of a circle,

which of allcircles approximates the profile,

most closely adjacent a given point. This definition is in harmony withthegcommon meanlng of the, term curvature radius or radius ofcurvature. v I

:In a profileior transmitting true uniform motion the'curvature radius atthe pitch point 21 equals the radius of-the aforesaid circular 70 errorof=the said character and magnitude pitch point.

points outside the pitch point are larger than the above said curvature radius of (R-sin a) andthey increase at a rate which is faster,

'the'larger' the distance of the considered pro- ;filepoint is from pitch point21. At a point 28 at the extreme tip of the tooth, and at a 7 point 29 situatedat the same distance from pitch point 21' adjacent the root of the tooth,

' the curvature radius is usually more than one and one half times the curvature radius (R-sin a) at pitch-point 21.

Preferably profile 27 is made a curve symmetrical with respect to normal 24 at a point (21) intermediate the profile ends, usually with respectto theprofile normal 24; at the -There exists, of course, one curve which is symmetrical with respect. to normal 24 at pitch point 21, and which transmits mathe i However a matically true uniform motion. Such curve might be described in relation with a coordinate system X, Y, of which the X 'axis is identical with the profile tangent atp'oint 21, and of which the Y axis is identical with normal 24. A pair of coordinates w, 3 corresponding to such a system are indicated'in Fig.2. large number of curves of the character described will have the same effect in practice, as the curve which is mathematically accurate. For it is understood that all that is-ne'cess'ary to fulfill themost exactthe ends of the active portion of the curve. 1

ing practical needs is to providea little modification of a circular arc, a modifiction which amounts to a few thousandths of an inch at 7 It is therefore sufficient to describe a few more characteristics of the curves which may beused. 1 Curves tangent to the X axis at the origm (21) of a coordinat esystem X, Y 'canvbe V mathematically described as follows;

in which (32,0 C C andso on are constants.

When the curve is fully symmetrical with respect to the Y axis, which is identicalto normal 24:,the constants C C G etc. are

zero. In accordance with the present in vention, at least constant C 18 zero, so that the above equation can be written: 7 V I y= 0 -w +O m -l- 7 should be negative and larger than C to fulfill exacting requirements. Another way of expressing this mathematical'statement is by 'sayin g' that profile 27 should extend notonly "outside of its curvature circle 18'but also out.-

f Small radii of curvature are known to be circle .18 (see; Fig. 2), and which can. be formed with simple mechanical means. i The Moreover it is found that constant" C side of the common par'abola'which contains the same curvature circle; The equation of the common parabola is known to be 3 02a,

whereas the ordinates y of profile27 are smaller, on account of the negative sign of constant C Fig. 3 further-illustrates profile 27 by-affording a comparison withan involute'pro-y file indicated'byd tted'lines -30." It 's clearly seen how'profile 27 extends outside of the in volute profile/on the tooth flank, andth'at'it provides "greater strength of-the tooth; The involute-proper ends "at its base circle 31, and contalns gradually reduced' 'radii of i curvature, the nearer a considered point of the involute-lles to'point 32, where the'involute reaches the base'circle 31. The curvature raf dius of the-involute, at point 32, is known to bezero. 1 ,7

a drawback by .causing' inuch-stressof the tooth surface when the teeth carry a-given load Mor e'over'they precipitate wear.

' Large i'adii of curvatureyasafforded by the present invention," reduce the stress of the contactingtooth surfaces. They givetherefore longer wear.

vif

. l Vear depends also on relativesliding: In

accordance; With the present,inventioni'the curvature radii of the basic tooth profilesin- '9 crease with increasing distance of theprofi'le points from the pitch point. Slidingalso increases with increa'sing distance ofcontacting' points from the pitch point. providing f larger radii of curvature and smaller sur face stresses atpoints which contain increased sliding, [the presentinvention also evens "up wear, that ist-o 'say it tends to make the wear moreuniformand-toconserve the exact tooth profiles.

tooth profile27 is tofirst assumea suitable symmetrical, curve of. the ,character described, whichextends outside of curvature latter requirement is most important Thereafter the profile is checked for errors. If the errors are not small enough, one may try another curve, for'there is a large choice of kinematica'l requirements. One such way of procedure will be described hereafter. v i Y A check which may be applied to. a curve is indicated in Fig. 3.

In gear pairs, of which each member is conjugateto the same basicpinion, as isthe case in the exampleshere described atlength, the profile 27 of the basic pinion is subject to the following requirement: 7

Take on pitch circle any two points 35, 36, which are equidistant from pitch further ims A practical wayofobtaining an'excellent point 21. Draw normals37, 38'tothe tooth" any two normals 37, 38 will intersect the pitch circle at equal angles-. On' the other hand the difference of the angles referred to is a measure of the departure fromthe theoretical profile.

Fig. 4 illustrates a pair of gears 40, 41 of which-both areconjugat'e to the same basic pinion. The latter contains tooth profiles of the character described with reference to-- Fig. 2 and Fig.3. In the instance illustrated gear4O is identical with said basic pinion and'c nta-ins tooth profiles 42, which are symmetrical withre spect to the'p'rofile normal 43 at the pitchpoint.

- at'saicl pitch point. The tooth profiles of faced grinding wh Th s ills indeed posh ing to said basic pinion.

gear-41' differ from the tooth profiles of gear 40, andmay be determined, if need be,

in the conventional manner from the profiles of the pinion. Both gears may be gen erated with a, gear shaper cutter correspond- Gear pair 40, 41 has a ratio between oneto one and two to one, such as is frequently used for instance in transmissions of automotive vehicles. Itisgthere desirable that the .tooth shape permits grinding with' a flat sible with gears formed in accordance with present invention; a'stvillj be further described hereafter; for also the toothjprofiles 44 of the larger gear 41 are wholly convex. They alsoj contain a minimum curvature,

radius at a point intermediate their ends; A toothprofile 44 is however not symmetrical with respect to the profile normal at its pitch point. Profile 44 endsadj acent the root with a'straight line element. In'other words the end 1 portion of tooth profile] 44, namely the portion which engages thetip of the fteeth of'gear 4Q,is notv curved at all, but contains contains adjacent its' f root endycurvature radii which gradually increase approachinfinity at point 45, "The straight line elefrom pitch point 50 to a point 52of the line of action. Let symbol e 'denote theangle" between" said radius [vector and the tooth normal" at pitch point 50. I Let further n denote the fturningfangle through which thebasic pinion would have'to turn tomove the tooth contact from pitch point 501a) point 52 of the line ofiaction; and let a denote the pressure angle as defined above. Angle F e can'ithenbe expressedasfollows', as will of teeth notexceeding-thirty.v

Gears of the character described and; formed in accordance with thepresent'in-f As explained, the curvature radii of said profiles are a-mi'ni num;

to a radius 47 drawn between'the gear centerbe understood by-those familiar with mathematicsy 1 i e=.E -u I E 'u +members of higher order... In this: equation the angles e and w are to be lntroducedin radians, as scom'mon in mathematics, and E IandE are constants.

In accordance with the present invention I preferably constant E -iskept between A;

cotan a and 4-00mm a; and'constani .E4.1S

negative and" larger than constantE The anu mber basic pinion preferably'contains ven'tion, and particularly gears with narrow 1 face may be finished after, hardening with a' or'plane working "surface '56;

may be rotated inany suitable known-mam. l

ner. A gear blank'58 is mounted ona slidef 59, which'is movable'iii :a direction perpen dicular to axis 60; ofgear' blank"5 8; along} guides providedon abed61. "jslid e 59 is con tinuously drawn in one direction by. means of a-wire or rope 65, to which a suitable l g i weight is attached. The gearblanl: given 1 a combinedrotary motion onits axis 60 and t n la 'yf o p pendiculaiito its axis; as isv quite usual in gear grinding, The de{ sired interrelation between' 's'aid rotary mo tion and'said translatory motion of thejblankff is effectedby acam dvice consisting of cam" 62 cooperating with aistationary element '63,?

which isangul'arlyaswell-as linearly adjust, able; During a grinding pass,-cam: 62-is1 maintainedin constantrelation to gearblank 5 8. The connection between the cam andgt s ing, as readily understood: "Gear flblanki158K. r l, may be'oscillated, and :th'e carn. .device jthen, imparts I a corresponding translatory motion to it, so that the gear 'fblank is moved pastabrading wheel55 in a ma1inerthat.atooth surface is successively engaged byfsaidabradQ ingwheel, In diagrammatic Fig. 5 a hand 'l lever 64 .isindicated as simplified meansgfor effecting oscillation of the flgear-blank'.

, The only novel feature of" the devicefindif cated in Fig. 5 resides in thefinterrelat'ion beQ tween the oscillatory motion and translatory motion of the blank',and in thecain device for. efiecting it.- For: thisQreason it was deemed... sufficient "to outline a very simple abrading.

device, and to enlarge new tions and'the'c'ain device; 1

A simplelcam devicepas maybe intaa abrading device of the characterf indicated in Fig. 5, Willfnow be separatelydescribed" 'ofr n i 0 tained 'inic'onst ant relationwithablank durl' J ing the pass of the blank past {an T ab rajding cam has position 66 indicated, in dotted Y gularity;and maintained "stationary during :Ithe abrasive process. I Let it be supposed at V .In Fig. agear blank having member 67 having a straight profile 68,

may be adjusted to any suitable anfirst that the orking surfiace ofthe abrading wheel is a plane parallel to line 68. It is V then ev ident to those tamiliar With mathematies-thatto produce a desired tooth confieur, th co to r m-. t be a e-m equidistant to said tooth contoun'that is to sey acurye obtainable fromthe tooth contour y P Q ga, stant d tanc 1 t e normals of said tooth contour and' by conec ing th Poin s s etermin gL,

e V a center is, indicated byitspitohcircle 71. p In another; position of the gear blank, here its nter-is poi 70 e d. t ;p ci n the position denoted .Withnumeral '71, the

lines.

. so desired, the "gear center may be given 7 in-additionia movement parallel to straight line,6 8, s0 that the gear center amo'vesf rom sa osition 7 0 to a position 7 3-orto a'position 3:. iSuoh additional motion will not ,jaifectthe angular position "of; the eamQand ito-nl v "Inthe em od e t lh mt d 6 the f gf ar, center is made to mov St shortens" or lengthens the distance covered by. the gear center.

line 'l';;illustrates anembOdiment,

in which the jgeqmenter' 70is made to'mox e on .a-oirele-M, in aclosed path.- em 66 enfiitlll rbl 7 Toothfp bfi 7755s e a ed e red n's h e 7Z8, h v g a fl plane vi orlging surface 78' parallel to straight line; 7 6. AnotherpbSition of the cam and p liifthefpositionflast referred to, the

V hgearteetli are out of they reach ofabrflf ingis disconnected. fromv cam 66 and indexed. The gear center (70, 705-) is moved at a con- 1 "WheeL'ZSQ In such a position thegear blank stant rate on its closed; circular path 74,:and

'abrading st artson a di'fi'erent tooth after the 9' 'g ari hasljcome within reaohfof rab'rading wh eze-1 l jGear. forming. processes so fan: described depend Vjerjjiinuch on the accuracy ioif reproduction of a computed form oftootli; or of {acam compntedin akicordancethe rewith. V Thus far it has been'disclosed howtheoretically' improved gears of high accuracy be U shall no proceed: to, I "show how the improved features .can be aetu e: llyend'e eim y fi i s prac is 1;

. blankl is indioatedibyrpositi on 70," 'ofthe Y cen terfand by a position 71 of the pitch menace-- other Words it will now be ei irpl in'ed shown, how-tooth for s-in accor ance ,W

the present invention,- and how suitableca useful in the r pro uct o by abras n, may; be obtained with livery simple 'mechanieal means, Which-permit accurate reprodu'ct at any time. a p e V a The first and main necessity is to beable to obtain the biasic form of tooth with simple, niechanicalnieansg The production of a pres;

ferred basic form of tooth, or of a-nqojrrep ng m pr fi w ll ew be xplain d- In Fi 8 numera ilf notes h nt r Of an element or gear containinga c rcular pro fil 1. en eredat'-p1 .i-nt.23;- BIQfilQ 18169 tacts with the straight-side of; an element; 81, which is movable in. the "direction of line 82. -Inanotherposition18 the circular ro 'file contacts with position 8.0 ofthe straight side of elements-81 It is evident that a plane "workingsurfacef ofan abrasiife Wheel-forms aciroular profile (18), ,when said abrasive wheel is moved; in,

the same manner as element 81 with respect,- toaa blank centered at point l4.

v In order to forma profile of the eliara oterv described and as indicated in dotted lines 83."

an ",li ne e e y p vide a ligh i correetion .of thetranslatory motion" of the abrasive Wheel or. of element 81 which may represent it. Profile'83'is seen to intersect side 80/ r element 8L .It is vident that element 81 should be moved a. little further further in such manner as to J Contact contin -v w t -1. 1 e d f di .0fi e18-3-v faced abrasive, wheel substituted for ele e t 1 7 A suitablenway of obtaining 'a correotion'. V

of he moti n of ele en jf fsiin l e e i Fig. 7 9;Elementl81 isfmaintained in; cone.

Link B an arm A. ate c nn ctedlp ith: e l j .v

e stantrlationwith pointf84 which isfcon-T j i tedy i k Baw an a m A" of a b k-Y 9 2 by 'ii t di e il poin zep I he; a

position 23, (if point 2 3,po int v84, has moiied' to mp si i n- 8&7, an side 8. fel mbent ,1. 113 moved to position 80",which is slightly" advanced in the direction of motion as compared With-thecorresponding, positionjn 1 F g-e 8- e u be he;

v turning angle '231 4f2 3 and let w denote'the angle between line 82 and link 13, 's hosew center; lineequals' line' 23Hf :.v i k V .1; 'l/Yith the above symbolsthe displacement;

.of 'p'oint 84 and of member 81 canbe methe;

m-atically expressed as, follows, as can; be!

matics p 7 v v Aina e o A: 1 "T e d spes m n hemed; wr -1.1; m m

e dily che j dby} anyQ l S i led} in; matheQ If on theiother hand element 81 i s moized V This expression characterizes the: difference differs from .a true: circular profile by as m a derived at... Fig. 9 by em loyin the customary. procedures. On account of angle to being small, the lfactor(1cos to) can be substituted by the'expression 'sin w.

With this substitution the term -B-(l cos 1.0) becomes:

of motion ofelement 81, in the motion 'defined by Fig. 9 and in the motion defined by F ig: 8, as effectedby a circular profile.

The expression is also a measure" of the difference between a circular profile and a profile as obtainable with a iiat faced abrading wheelinmotions corresponding to 9.

It should be notedqthat the above. expres sion is of the same character. asthe. expression derived for thetotalerrorE of-truly circular tooth profiles, the. variable to. appe'ar-ingm:

the same form (17005 u) -'in both cases. Itistherefore,understoodthatthe profile obtainable with motionsas indicated in Fig. 9

amounts, whichmay: be selectedto just make up for the said computed'error of the circular One feature shouldbe particularly noted, Q namely the change of theratio between the.

rotary motion ofrarm Aand the translatory motion of point 84 and element81. :It can be readily. demonstrated that in any position they instantaneous ratio between the traI'1slatorymotion and the-rotary motion is given by a changing leverage (L), which in Fig. 19 is equal to the distance between point 14 and the intersection point between line 14-23 and line 28".8l." 'In; other words an element of the translatory, motion. equals an element; of .the rotary said-leverage (L).

nm tiplied by In the central P n of tl edvice 9,

the said leverage (L) is equalflto the length or arm A. In any angular position of arm A. adjacent said central position the said leverage (L) is smaller than the length of armA,'. as .is readily seen. 1 Translation is. therefore i' efit'ected at a changing rate, which is a maxi mum i'nthe central position shownin Fig. 9.

' Similarly, tooth profiles of the character described may beformedrin aiprocess small angle defined by used which the blank performs a translatory and;

a rotary motion, whereby the translatory ma tion contains a maximum ratio as compared with the rotary. motion, when an intermediate point of the tooth profile is being formed.

In principle it does not make any difier'ence whichmotion is performed'ibyatool. and

which motion is performedby theblank, as long .as the relativeinotions are the same.

"which is maintained in unison with armsA durmg the abrasive passes. A.'flat faced gear blank. IThe-plane working surface'of the abrading-wheel is herepe'rpendicularto' the direction of motion of the blank, that is to say to line 86.

The same linkage-A with respect'to Fig. 9 connects the gearblank with apoint 84, which is maintained inconstant relation to the working face ofabrasive wheel 87 and which is here stationary. The

center of the blank is shown in a further positionl i, to which corresponds aposition23. of the joint between arm A and link B.

It can be-readily demonstrated, thatin the" arrangement illustrated by Fig. 10 the fol symbols are kept: f a I J I 2 AFR-cos... a .Moreover, i 1

or. between (A-tan and i pending on. the efi'ectdesiredi. It is often desirable to modify the toothprofiles slightly; f

adj acent'. their ends, as is well known. I

In the arrangement indicated in; Fig. 11,

the vplane working face of abrasive wheel87 'is set to an angle such as the above. defined angle a. In this case -armAequals radius R. To obtain the same efl'eot-as in'the arrange In Fig. 10, and in the following .figui'es,it is supposedthat a translatory motion in direc- 1 i tion of line 86 as well as a rotary motion on center a, 14 are both imparted to the blank,

and B as explained abrasive wheel '87 is shown in engagement. I with the pitch portion-ofthe tooth 88 of a lowing relations exist, when the above defined j r iio ment. shown in 'Fig. 10,- length B is selected it 7 in the same proportion to length A as th ere. B is preferably selected between A'tan a f and 2A tan a.

The same relations-betweenlk and Bmay also be used for forming master cams'cor responding to the basic-form o'f tooth, a s the profile of such camsis simply'aline'equidistant to a tooth profile,'which' requires the same generating .motions or forming motions I as the tooth profile.

The 'motions described with reference to tainedwithv eccentrics or cams of form, as will now be described.

a In Fig. 12 alinkage of thefchara'cte rijusti:

Figures 9 toll can be obtained by providing actual linkage, or1preferably; they are obcircular described is indicated by auniliary lines Ap andB. .Theturning; center of a blank is shown in. a position 14:, and the oint which of the aforesaid linkage; I

connects arm A withlink B in a position23. The means used for effecting the position prescribed by the. linkage are a stationary circular element-91, or roller 91, and acam Orv-eccentric 92.= Element 91 has a fixed center 84,.whi'ch coincides with thestationary point r The circular profile of carn-92wis centered at point'23". On accountof thecircular pro "filescentered at pointsS i and 23 respective-lyyit: is evident that point 23Viskept at a constant'distances from point 84", provided .onlyzthat the said circular profilesare main:

taiiied i'n contact with each other The said 7 distanceis: equalf'to' the sum: of the profile radii, asisreadily understood- Furthermore, point23fis. kept at aconstant distance from turning center 14",as said centeria lso forms thetturning oenter'of'cam 92.

.Itis therefore seen that thea.circular cam.

' 92% and the stationary element 91 constitute very simple means-which may accurately replace the aforesaid linkage. 7

Ehe' central position 92' of cam is indicated in dotted lines in- Fig. 12;

- Fig. 13" and Fig. 14. illustrateaslightly modified embodiment for obtaining the dc sired motions by means of circular cams cooperating with circular elements. The links. age :which prescribes themotions. is precisely the same as'in Fig. 12.. The means use'dfor efiectin'g'the motions differ notv only in the use of an adjustable cam 93, butalso in the 7 use of a concave profile 94 on the stationary element,whichcooperates with the circular ,profilewof cam Profile9l is; a circular are-centered atpoint 84; Cam 93 is adjustable" readily along. straight guides 95 of an armrigidlysecured to shaft 96,:to which also the blank is secured. It is-noted thatitheprovision of a co-ncaveprofilei9t permits to obtain cam: surfaces. of large diameter and to.

reducev surface stressesand wear through the ,camacontact. The fact-that theprofiles used ontcarni93 and on the stationary element are circular, and broadly are symmetrical curves,

, facilitatesvery much their accurateprod'uc tion.t3 If so desired, a circular roller centered. at point 23" may be used in place of circularf- 'c'a-nr engaging an. adjustable: stationary ele' cam..93. In thefollowing and inthe claims rollersare included in the broad term'camn Fig. 15iand Fig. 16 .illustrate an adjustable ment,.suchras is usefuhfor finishing byabra present invention.

siongearsconstructed in accordance with the Cam98 may be adjusted; radiallyof 9,9..alongxtapered sides .100 of an arm 101 which is integralwith shaft 99;. 'A-screw' 103. servesito clamp cam '98 to arm :1O1Linany.

suitableposition'. cam 98 contains acircui' lap profile; and engages, a. hollow stationary element 194; The cam surface 105 is a cfoni;

cal surface. engaging. a hollowfconical'sur face 1060f element 104.,E1ement 104 is secured to a holder 108, which may be adjusted alongafbase:109f. Element 1'04: may fur,-

thermore be adjusted upwardly-and downs ,wardly on holder. 108, and fixed in any desired position after introducing a spacing block- 110- of suitable thickness. Element 105 may further be adjusted on ho1der'198inithe direction of shaft 99. A suitable spacer 111 is usedgto obtain a desired position lengthwise of shaft 99. t The last said adjustment, together with anadf'justment ofho-lder 108, serves tochangetherelation between the: par-. allelcenterllines of the conical surfaces 105'- and 1306'. In otherworlds it serves to change the aforesaid distance B.

g A cam device, having adjustmentsasirrdicated in Fig. 15 and F ig; 16 may be used in forming byabrasi-on' gearsformed in accordance with the present invention, as well as othergears'andc'ams.

Fig. 17 points out aLca-m: arrangement for obtainingrbasic tooth forms and master cams 1n a modified way? Cain 1 121 is secured to shaft1l-3 to whichalso a blank is secured.

Cam: .112 contains ac-hollow" circular profile 1141, which engages a convex stationary ele menial-1'15... vlf'nrtheinstance ilhistrated thecom' vex profile 1:16 of elementv 1-15 is? an involute havingabasecircle 117.

.:.Once the basic tooth form is mechanically";

determined, otherutooth forms may' 'be'iob tained for instance through;- generation' with" r a pinion shaped cutter containing said basic tooth forms linithis'operati on the-'cutteris" reciproca-ted and-the cutter and blank rollupon: each other in; the manner of two inter j meshing gears. f It: is of course understood that a pinion toflbe used as a cutting: tool should be 1, provided with suitable relief, so 7 Y that only the cutting edges: engage-a blank. Such relief may. befprovided known" 1 g I manner. i

Toothforms ofggearpairs of'coiiiparativea' Y ly-smalzlxratiosmayfurthermoreibe directly obtained with an abrasive deviceof: other wise zknown character, byintrodiucing there an ad ustable cam; device of the character described: with-respect to Figuresll to 16,

and by suitably adjusting said cam device;

Fig. 18.- illustrates a. cam device set for Y the:

generation ofagear of larger diameter'than a basic pinion of the character describedin detail; and; Fig: '19 illustrates; a. device set for; the generation of a: gear of: smaller f diameter xthanfthe basic pinion". The pitch,

circles of thealrespectiye gears are denoted to the aforesaid angle a and in the indicated position of the gear blank, contact between the abrasive wheel and said gear blank is made at the pitch point of the tooth profile of the gear blank.

To formgears constructed in accordance with a preferred embodiment-of the present invention, the ratio between the translatory motion of the gear blank and its rotation should be a maximum whenabrasive contact is made at the pitch point ofa tooth profile of the gear blank, as hereinbefore explained. In other words the leverage (L) which characterizes said ratio should be a maximum when grinding contact is made at the pitch point of a toothprofile, and it should be re: duced in adjacent positions of the gear blank, at a rate which will now be described.

For convenience, let R denote the pitch radius of a gear constructed in accordance with a preferred embodiment of the present invention, and let u denote the turning an gle of said gear as compared with the angular position of thegear in which abrasive contact is made at the pitch point of a tooth profile. Further let as herein before R denote the pitch radius of a basic pinion characterized by having a minimum radius of 4 curvature at the pitch point of its tooth profile; and let u denote the turning angle of said basic pinion corresponding to turning angle 10' of the aforesaid gear. In other words let u be defined as follows:

I u u R M It can be demonstrated with the known methods of mathematics that to obtain the tooth characteristics explained, the leverage L should be reduced with increasing angle a or u as follows, a being introduced in radians (as the length of an are measured on a circle of a radius one) wherein the dots indicate members of a higher order in u, for instance members containing the factor if, or w, or u etc.

Attention should be focussed especially on the coefiicient of the member containing a", which characterizes the rate at which the leverage L is reduced at small angles a and u,

and alsoin the average. Moreover itis found that leverage L is reducedmore rapidly when grinding the flank portion of a tooth profile than when grinding its tip portion. Further characteristics may be computed with the known methods of mathematics with a the directions given.

in Fig. 19 numeral 130 denotes the intersection point between the elongation 131 of arm A and a line 132, which is drawn through stationary point 133 perpendicular to the direction 134 of motion of center 135 of the gear blank. Point 136 isthe interwith theuse of a scribed.

section between line 137133 and a line drawn through gear center 135 perpendicu Line 138 is drawn through lar to line 134. point 136 perpendicular to line 137-133. It should be noted thatline 138- passes through the aforesaid point 130. It can be demonstrated mathematically that this characteristic expresses the requirement that the a ratio between the .translatory motion and the rotary motion of the gear blank. is a maximum in the position of the cam device as indicated in full lines.

Other positions are indicated'in Figi. 18

and Fig. 19 by the position of the linkage only. M j 7 It is found that gear pairs ofcomparatively small ratio may be for-med very'accurately simple cam device as de While in the exampleschosen for explanation spur gears were particularlyreferred to, it is understood that theinvention is not confined to spur gears, but is equally .appli-' cable to gears mounted on parallel axes and to gears mounted on intersecting:axes,' that is to sayto gears whose pitch surfaces roll on each other without sliding, and also to other gears; The teeth may be straight or curved and extend along. generatrices'jof the pitch surfaces or at any angles" thereto. Moreover the invention is not confined to gears, but is applicable broadly to gear l shaped article's, including particularly gear rality of equal tooth like projections, the side profiles of said projections being wholly convex and having a minimum radius of curvature at a point intermediate the ends of a profile, said profile being substantially symmetrical with respect to the profile normal at said point. I 1 v a.

2. A gear shaped article, containing a pluioo rality of equal tooth like projections, the side surfaces of said projections being so shaped as to contact along a line with a plane tangent to a side surface, the profile of a side surface being wholly convex and having a minimum radius of curvature at a point intermediate the ends of a profile, said profile being'sub-,

stantially symmetrical with respect to the profile normal atsaid point.

3. A gear shaped article, contain ng; less than twenty five equal toothlike pro ect1ons, the side profiles of said projections being wholly convex and having a minimum radius of curvature at a point intermediate the ends of'a profile, said profile being symmetrical with respect to the profile normal at said point.

4;; A gear shaped article,containing a plurality of equal tooth like projections having convex side profiles in a plane perpendicular to the-axis of sa1d article, sa1d profiles contaming a minimum curvature radius at a mean profilepoint, the curvature radius at the root ends of said profiles being at least fifty percentlarger than said minimum curvature radius.

7 cent larger than said minimum curvature radius.

5.]A gear'having'fa convex tooth profile havingxconve-x active'tooth profiles, said activetooth profiles having a minimum curva ture radiusat their pitch points.

8. A gear of lessthan twenty five teeth, having convex active tooth profiles, said active tooth profiles having'ami'nimum curvature radius at a point intermediate, their ends, the curvature radius at the root end of an active tooth profile being at least fifty per- 9. A pair of gears whose pitchsurfaces roll on each.v otherwithout sliding, containing a lineof action of continuously and gradually changing curvature, said line of action hav ing a point of inflection adjacent its pitch point and corresponding to an equation e=E -u l-E- -u +E -u +members of higher order 1n u,,where1n c and u are angles measured 1n radians,,e denoting the, angle included: by the tangent to the line of action at said point of inflection with a radius drawn from said point of inflection to a point of the line of action, u referring to the smaller gear ofsaid pair and denoting its turning angle which corresponds to the gear action; between said point of. inflection and said point of the line of action, and wherein E E E are constants, E being larger thanv one.

10. Apair of gears having straight or helical teeth, containing a line of action of continuously and gradually changing curvature on. its whole length, said line of action- .extendingin'polar symmetry with respect to its pitch point in such manner as to furnish a minimum pressure angle at said pi ch point. -11. A pair of gears having straight or helical teeth, containing a line of action of continuously and gradually changing curvature, said line of action having a pointof inflection at its pitch point and extending in polar symmetry with: respect to said pitch point, s

as to correspond to an equation e=E -u +E -u*+ ing the angle included by the tangent to the v line of action at said pitch point with a rad-i us drawn from said pitch point to a point of the line of action, a referring tothe smaller gear of said pair and d'enotingits turning anglewhich corresponds to the'gear action the-lineof action, and wherein E E are constants, of which E; is negative and more than twice as large as E w 12. A pairof gears of whicheach member is conjugate to the same basic pinion said basic pinion having a tooth: profile which is syimnetrical with respect to the profilenormal at the pitch point of said profile.

18'. In a pair 0t gears of a ratiobetween two to one and one to one, a larger gear hav ing convex active tooth profiles ending with a straight line element adjacent the roots of the teeth, said straightline element beingbetween said pitch point and said point of disposed at an angle to a radius drawn through the gear center, and a smaller gear having convex active toothprofil es containing a minimum radius of curvature at a point intermediate the ends of saidgprofiles, f

14. A pair of gears mounted on parallel axes, each gear of said pair beingconjngate to the same basic pinion, said'hasic pinion having a convexjtooth profile with a minimum curvature radius at its pitch'point, and said profile being symmetrical with respect to the profile normal at said pitch point.

15. A pair of gears mounted on parallel axes, each gear of said pair being conjugate to a basic pinion, said pinion having a convex tooth profile with a minimum curvature radius at a point intermediate the profile ends, and sa1d profile being symmetrical with respect to the profile normal at sa1d point.

ERNEST WILDHABER. 

